On spontaneous fission and alpha-decay half-lives of atomic nuclei

0.42 MB·English

by  K. Pomorski

On spontaneous fission and α-decay half-lives of
atomic nuclei
5
K Pomorski, M Warda, A Zdeb
1
0 DepartmentofTheoretical Physics,MariaCurie-Sk lodowskaUniversity,Lublin,
2 Poland
E-mail: [email protected]
n
a
J November2015

6
1 Abstract. It is shown, that the Gamow-like model with only one adjustable
parameter – radius constant is able to reproduce well the alpha-decay half-lives

] foralleven-even nuclei withtheproton numberlargerthan 50. Thesystematics
h forodd-Aandodd-oddisotopescanbealsowelldescribedwhenonesintroducesan
t additionalhindrancefactor. SimilarmodelbasedontheW.J.S´wia¸teckiideafrom
-
l 1955isdeveloped toreproducethespontaneous fissionhalf-livesoftransactinide
c nuclei. The achieved accuracy of reproduction of the data is better than that
u obtainedinmoreadvanced theories.
n
[

1
PACSnumbers: 21.10.Dr,21.10.Tg,23.60.+e
v
2
1
9 Keywords: nuclearfission,alpha-decay,fissionbarrierheight,half-lives,Gamowmodel,
3 liquid drop model
0
.
1
0 Submitted to: Phys. Scr.
5
1
:
v
i
X
r
a
On spontaneous fission and α-decay half-lives of atomic nuclei 2
1. Introduction ofdeformednucleiwasdevelopedandtheIngliscrank-
ingmodelwasimplementedtoevaluatetheinertiacor-
Large progress in synthesis of heavy nuclei done in respondingtothefissionmode. Ofcourseintimethese
the last decades raises new demands for a better and modelsgivemoreandmoreprecisereproductionofthe
better theoretical description of their decay modes. spontaneous fission half-lives systematics and the fis-
Spontaneous fission and α-radioactivity are the most sion mass distributions. A short critical description of
important processes of disintegration of heavy nuclei. the spontaneous fission theories will be presented in
Alphadecayoccursmostofteninactinidesregion, ourpaperinordertobetter understandphysicalback-
but is also observed in isotopes with Z ≥ 52. The ground of a simple model for the probability of this
first theoretical interpretation of alpha-decay process decaymode,whichwehavedevelopedfollowinganold
was given independently by Gamow [1], Gurney and idea of W. J. S´wia¸tecki. Namely, in 1955 year he pro-
Condon [2] in 1928 year. Emission of alpha particle posed a formula which was able to describe the global
is treated as quantum-mechanical tunnelling through systematics of the spontaneous fission half-lives [9].
the nuclear Coulomb barrier, where the probability His simple phenomenological formula, based on cor-
of emission is calculated using one-dimensional WKB relations between logarithms of observed spontaneous
approximation. Recently it was proven, that this fission half-lives and ground state microscopic correc-
approximationcanbesuccessfullyusedtoevaluatethe tions, reproduced well the experimental data known
probability of tunnelling by alpha particle and cluster at that time. We are going to show in the follow-
aswell(seeRef.[3]). Itwasalsoshown,thatwithinthis ing that his idea, combined with the modern version
simple model (containing only 1 adjustable parameter of the liquid drop model (LSD) [10], allows to obtain
for even-even nuclei) one can reproduce alpha decay satisfactory agreement with the measured up to now
half-lives of heavy emitters with higher accuracy in spontaneous fission half-lives.
comparison with modern (containing 5 parameters) The paper is organized as follows: the main
version [4] of Viola-Seaborg formula [5]. assumptionsofα-decaymodelandresultsobtainedfor
The observation in 1938 of the neutron induced emitters with 52 ≤ Z ≤ 110 are presented in Sec. 2.
nuclear fission by Hahn and Strassmann came rather The WKB theory of the fission barrier penetration as
unexpected [6]. This new phenomenon was explained well as the the semi-empirical S´wia¸tecki’s formula for
within a few weeks by Meitner and Frisch’s who spontaneous fission half-lives will be described in Sec.
established the most important features of low-energy 3,whereresultsfortheisotopeswith90≤Z ≤114are
fission: the energy released in this process is equal analyzed. Sec. 4. contains Summary.
almost 200 MeV and results from the Coulomb
repulsion of the fission fragments and the number
2. Alpha decay
of neutron emitted per fission event larger than one,
what has opened a possibility for a chain reaction [7]. 2.1. The model
One and a half year after Hahn and Strassmann’s
discovery, Flerov and Petrzhak have first observed The quantum tunnelling theory of alpha emission
the spontaneous fission of Uranium [8]. As we have assumes, that the decay constant λ is proportional
written above,Gamow had explained the α-decay and to the barrier penetration probability P, frequency of
its sometimes rather long half-lives by a quantum assaults on the nuclear Coulomb barrier per time-unit
tunnelling process of a preformed α particle through nandparticlepreformationfactorSα. Inthepresented
a Coulomb barrier. So, according to the concepts model the expression for decay constant is simplified
of Meitner and Frisch, one could expect spontaneous as the effect of the preformation is included into the
fission of uranium also from the ground-state, but probability P (see discussion in Ref. [3]):
with a considerably longer half-life than for the α- λ=nP . (1)
decay, because of the larger reduced mass for almost
The probability P of tunnelling through the barrier
symmetric fission.
is calculated using one-dimensional WKB approxima-
The first quantitative estimates of the sponta-
tion:
neous fission probability becomes possible, when the 2 b
microscopic-macroscopicmodelofthepotentialenergy P =exp − 2µ(V(r)−E )dr , (2)
α
¯h
(cid:20) ZR (cid:21)
p
On spontaneous fission and α-decay half-lives of atomic nuclei 3
2 2
Te Dy Tl Rn
0 0
-2 -2
2 2
I Ho Pb Fr
0 0
) -2 ) -2
p 2 p 2
exT1/ 2 exT1/ 2
cal / 1/2 0 Xe Er cal / 1/2 0 Bi Ra
T T
( 0-2 ( 0 -2
1 1
g g
lo 2 lo 2
Sm Tm Po Ac
0 0
-2 -2
2 2
Gd Yb At Th
0 0
-2 -2
50 60 70 80 90 60 70 80 90 100 110 120 130 100 110 120 130 140
N N N N
2 2
Lu Os Pa Cm
0 0
-2 -2
2 2
Hf Ir U Bk
0 0
) -2 ) -2
p 2 p 2
exT1/ 2 exT1/ 2
cal / 1/2 0 Ta Pt cal / 1/2 0 Np Cf
T T
( 0 -2 ( 0 -2
1 1
g g
lo 2 lo 2
W Au Pu Fm
0 0
-2 -2
2 2
Re Hg Am Md
0 0
-2 -2
70 80 90 100 90 100 110 120 130 140 150 140 150 160 170
N N N N
Figure 1. Differences of the theoretical and experimental [11] Figure 2. The same as in Fig. 1, but for isotopes with
alphadecayhalf-livesonlogarithmicscale,calculated fornuclei 81≤Z≤90(top)and91≤Z≤101(bottom).
with52≤Z≤70(top) and71≤Z≤80(bottom).
On spontaneous fission and α-decay half-lives of atomic nuclei 4
differs from that reported in Ref. [3], as now more
alpha emitters were taken and the cluster decays were
2
not included in our analysis. Fitting procedures of
No Sg
0 hindrance factor were performed for odd systems with
fixed r value (h = 0.25 for odd-A and doubled for
0
-2 odd-odd emitters).
2
) Lr Bh 2.2. Results
exp 1/ 2 0 The logarithms of the ratios of the half-lives (Tcal),
T 1/2
calT / 1/2 -22 c(Tal1ec/xu2pl)atfeodruasllinegxaabmoivneedfornmucalleisima,retosthhoewmneiansuFriegds.on1e-3s
( 0 Rf Hs as a function of the neutron number N.
g1 0 Large deviations from the data are observed for
lo emitters with neutron number N = 127 in 83Bi
-2 - U elements (Fig. 2), where underestimations of
92
the log (T ) reach about two orders of magnitude.
2 10 1/2
Calculated half-lives of some lighter emitters ( Te -
Db Ds 52
0 74W)areoverestimated,butthesediscrepanciesdonot
exceedone orderof magnitude. The root-mean-square
-2 Mt deviations of our estimations made for the α-decay
half-lives for all considered nuclei are summarized in
140 150 160 170 140 150 160 170
Table 1.
N N
Figure 3. The same as in Fig. 1, but for isotopes with Table 1. Root-mean-square deviations of log (Tα ) calcu-
10 1/2
102≤Z≤110. latedusingEq. (6)withtheradiusconstantr0=1.23fm.
where µ is a reduced mass, R is the spherical square
π −π n h r.m.s.
Z N
well radius
e-e 127 0 0.39
R=r0(A1α/3+A1d/3) (3) e-o 97 0.25 0.66
and b is the exit point from the Coulomb barrier: o-e 82 0.25 0.54
o-o 54 0.5 0.79
Z Z e2
α d
b= . (4)
E
α
A ,Z and A ,Z are the mass and proton numbers
α α d d 3. Spontaneous fission within the WKB theory
ofanalphaparticle(orcluster)andadaughternucleus
respectively. E is the kinetic energy of emitted alpha
α A continuous interest in the theoretical description
(cluster) particle [11]. The number of assaults per
of the fission dynamics is observed since discovery of
time-unitnisevaluatedfromthe quantum-mechanical
this phenomenon in 1938. In present Section we are
ground-state frequency in the spherical square well:
going to concentrate on the spontaneous fission and
π¯h its description within the WKB theory only. The
n= . (5)
2µR2 results presentedhere will be used to understand, why
Inthisformalismtheα-decayhalf-lifecanbeexpressed a simple one parameter (for even-even nuclei) model
as follows: `a la S´wia¸tecki [9], described in Section 4, is able
to reproduce the spontaneous fission half-lives of all
ln2
T1/2 = λ 10h , (6) knownnucleiwithhigheraccuracythanobtainedusing
more advanced theories.
where constant h (so-called hindrance factor) was
Quantum mechanically it is possible for a nucleus
additionally introduced for odd nuclei. The least-
in its ground state to tunnel the fission barrier.
square fit of the radius constant r was performed to
0 The probability of the barrier penetration depends
known experimental half-lives of even-even α-emitters
not only on the height and width of the barrier
(127 cases) [11]. Only the most probable alpha decays
but also on the magnitude of the collective inertia
mode at each isotope was chosen in our analysis.
associated with the fission mode. The calculations of
Obtained in this way value of the nuclear well radius
the spontaneous fission half-lives T require a careful
constant is equal to r = 1.23 fm, and slightly sf
0
On spontaneous fission and α-decay half-lives of atomic nuclei 5
evaluationofthecollectivepotentialenergysurfaceand
6
the collective inertia tensor. Different models, like the static
4 dynamic
macroscopic-microscopicmodel[12,13]ortheHartree-
250
Fock-Bogolubov self-consistent theories (see e.g. [14]), 2 Fm
can be used to generate the potential energy surfaces. 0
The collective mass parameters one evaluates usually V]
Me -2
in the cranking approximation [15, 13] or within V [
the Generator Coordinate Method (GCM) with the -4
generalized Gaussian Overlap Approximation (GOA) -6
[16]. E=Veq+0.5 MeV
-8
0.08 ∗ -10
V( ε , ε ) 0.8 1 1.2 1.4 1.6
4
s
[MeV]
10 1200
0.04 static
static
10 6 2 6 6 dynamic 4 1000 phenomednynoalomgiicc
∗ε4 2 irrotational
0 800
V]
252 e
Fm M
2/ 600
−0.04 [/hss
B 400
10
14
200
−0.08
µ
0 0.2 0.4 ε 0.6 0.8 1.0 0
0.9 1 1.1 1.2 1.3 1.4 1.5
s
Figure 4. A contour plot of the energy surface for 252Fm is
showninthetwo-dimensional quadrupole(ǫ)andhexadecapole
(ǫ4) deformation parameter space. The dashed lineis the path Figure 5. Collective potential (top) and inertiaBeff (bottom)
of steepest descent from the saddle points to the ground state along the static (thick solid line) and the dynamic (thin solid
andtothescissionline,thethicklineistheminimal-actionpath. line)pathtofissionasafunctionoftherelativedistancebetween
AfterRef.[17]. the fragment mass centres s = R12/R0. The reduced (µ),
irrotational flow and phenomenological inertias [19] are shown
forcomparison. AfterRef.[20].
The spontaneous fission half–life is given by:
ln2
Tsf = . (7) Determination of the fission path in the deforma-
1/2 nP
tion space plays crucial role during evaluation of the
Here n is the number of assaults per time unit,
fission probability [13]. The action integral (9) should
associated with the frequency of zero–point vibration
be calculated along such a path (so-called dynamic),
ofthenucleusinthefissionmodedirection. Thefission
which minimize its total value, thereby maximize fis-
barrier penetration probability P might be evaluated
sionprobability(8)[18]. Incontrastto the static path,
withintheone-dimensionalWKBapproximation,what
thedynamicone(dependentontheinertiatensor)does
leads to the following expression:
not haveto leadthroughthe bottom of the fissionval-
1
P = . (8) ley,asonecanseeinFig.4. Thecomparisonofthefis-
1+exp{2S(L)}
sion barrier and effective inertia, corresponding to the
The action–integral S(L), calculated along a fission dynamic and static fissionpaths for 250Fm is shownin
path L(s) in the multi–dimensional space of collective Fig.5. The collectivepotential V(s), correspondingto
coordinates is given by: the dynamic path, is larger than in case of the static
s2 2 1/2 one, but the dynamic inertia is smaller. It might be
S(L)= B (s)[V(s)−E ] ds , (9)
¯h2 eff gs roughly approximated by the so-called phenomenolog-
Zs1 (cid:26) (cid:27) ical inertia fitted to the observed spontaneous fission
where the integration limits s and s , correspond to
1 2
half-lives [19]:
the classical turning points. E is the energy of a
gs
nucleus in its ground state and V(s) is the collective 17 3
Mphen =µ 1+k exp −a(R − ) . (11)
potential. Bss(s) is an effective inertia tensor in the R12R12 15 12 4
(cid:18) (cid:20) (cid:21)(cid:19)
multi-dimensional space of collective coordinates {q }:
i Here µ is the reduced mass, corresponding to the
dq dq
Bss(s)= Bkl k l. (10) relative distance between fragments (R12=3/4 for a
ds ds sphere) and a = 2.452 is a numerical constant,
k,l
X
On spontaneous fission and α-decay half-lives of atomic nuclei 6
20 theory quadrupole phonon energy, which is about 1 MeV for
exp. actinides. Several authors even treat ∆E as a free
parameter. Itshouldbestressed,thattheHFBtheory,
U
15
as each variational method, gives the upper limit of
the ground-state energy, so the zero-point correction
Pu
10 energyatthe equilibriumpointisapproximatelyequal
Cm to the difference of the ground-state and the potential
energy at the minimum (see Ref. [23]). It can be
) 5

y] easily shown, that the zero-point energy of harmonic
F [2 Cf Cm oscillations is equal to the half of the corresponding
ST1/ 0 No phonon energy, i.e. (21¯hω) [24], what is consistent
( with result, obtained in Ref. [25], where the coupled

og -5 Rf quadrupole and octupole vibrations in the region of
l Ra–Th nuclei were analyzed. This conclusion is also
valid even for strongly anharmonic motion such as
-10 Rf
Fm pairing vibrations [26].
The number (n) of the collective degrees of
Ds
-15 freedom results much on the value of the zero-point
Sg Hs
energy correction. Namely, the ε is proportional to
0
-20 n·¯hω. Thus,whentheHFBequationsaresolvedinthe
wholen-dimensionalspaceofcollectivecoordinates,all
134 140 146 152 158 164 zero-point corrections should be substracted from the
N energy to obtain the collective potential:
V(s)=E (s)−εfiss(s) , (12)
Figure 6. Spontaneous fission half-lives of even-even HFB 0
transactinides and their shape isomers (II) obtained with the whereεfissisthezero-pointenergyrelatedtothefission
0
WKB approximation (8) along the least action (dynamical
mode [16].
trajectory). AfterRef.[18].
In the case of the one dimensional fission barrier
penetration given by the Eq. (9), the oscillations
determined from a fit to the exact irrotational flow perpendicular to the fission path are not included
inertia[21]fork =1. ItwasfoundinRef.[19]thatthe during evaluation of tunnelling probability (16).
phenomenological inertia obtained with the constant In this direction the zero-point energy corrections
k =11.5reproduces well the systematics of the known will be approximately cancelled by the 1/2 of the
spontaneous fission half-lives. corresponding phonon energy.
Spontaneous fission half-lives of even-even trans- In Fig. 7 the potential and the binding energy
actinides evaluatedinRef. [18] arepresentedin Fig.6. alongthe fissionpath areplotted. As one canobserve,
The estimates correspond to the least-action trajecto- the zero-point corrections may affect on the fission
ries (dynamic path to fission). The WKB approxima- barrier height.
tion(8)andthecrankinginertiatensorwereused. The
two-dimensionalpotentialenergysurfacesforeachiso-
E
topewereobtainedwithinthemacroscopic-microscopic HFB
model using the Nilsson potential and the droplet en-
εfiss
ergy for the macroscopic part. The effects of the axial 0
asymmetry as well as the reflection asymmetry were
V
taken into account.
E
Usually calculations of spontaneous fission half- gs
lives are performed with an assumption, that the
collective potential is equal to the Hartree-Fock-
Bogolubov (HFB) binding energy or its macroscopic- s1 s2 s
microscopic approximation. The zero-point energy
corrections are often neglected in papers dealing with
similar problems. We shell discuss this point later.
Figure 7. Collective potential V(s), binding energy EHFB(s)
Moreover,oneassumes,thatthegroundstateislocated andground-stateenergyEgs ofanucleus alongthefissionpath
at ∆E ≈ 0.5 MeV above the minimum of the HFB s. AfterRef.[20].
potential. As a justification of such a choice of
∆E one refers to experimental values of the average The determination of the least action path in
On spontaneous fission and α-decay half-lives of atomic nuclei 7
deformationspaceisnotatrivialtask(seeRef.[13]for descriptions of the spontaneous cluster decay. One
furtherdiscussion). Theapproach,whichisworthtobe can use a fission-like mechanism to reproduce the
mentioned in this context is dynamical programming main features of the process [28, 31, 32, 33, 30] or
method [17, 18]. Nevertheless, applying this method one assumes, in close analogy to the α-decay, that a
one should choose optimal distance between the mesh cluster,performedbyanonadiabaticmechanisminside
pointstoavoidlargenumericalerrors. Thatisbecause nucleus, penetrates the potential barrier created by
the accuracy of calculations of the partial derivatives the Coulomb and nuclear interaction with a daughter
∂qi,whichresultmuchontheeffectivemassparameter nucleus[34,35,36,37]. Thepotentialenergylandscape
∂s
(10) as well as the zero-point energy correction (see of 230Th obtained by the HFB calculation with the
e.g. Ref. [16]) along the fission path, strongly depends GognyD1Sforce[30]ispresentedinFig.8asafunction
on the grid size. The von Ritz implementation (see of quadrupole (Q ) and octupole (Q ) moments. Two
2 3
appendixofRef.[27])mightbeusefulasanalternative valleys leading to the cluster emission (CE, solid line)
for practical calculations of the spontaneous fission and to the asymmetric fission (AF, dashed line) are
probability. Themainadvantageofthismethodisthat visible in the plot.
the influence of the zero-point energy correction along The above theoretical models, which describe the
thedynamicalpathsmighteasybetakenintoaccount. cluster-radioactivity, are rather complex and contain
Typicallythemassyieldofthespontaneousfission adjustable parameters. Last year we have shown
fragments of the actinide nuclei is spread between in Ref. [3], that within a simple Gamow model
70 and 180. Usually, except the bimodal fission of with only one adjustable parameter (radius constant)
258Fmandneighbouringnuclei,the asymmetricfission common for the α-decay and the cluster emission one
ismoreprobablethanthesymmetriconeandthemost can describe with good accuracy the experimental
populated mass of the heavierfragmentis around140. systematics of half-lives for the cluster radioactivity
A very mass asymmetric spontaneous fission process, of even-even nuclei. An additional hindrance constant
when the mass of the lighter fragment is around 20, is was introduced in Ref. [3] to describe the cluster
already considered as a cluster emission. This type of emissionprobabilityfromodd-even,even-oddandodd-
radioactivitywaspredicted in 1980by Sandulescuand odd isotopes.
co-workers [28] and four years later it was discovered
by RoseandJones[29],whoobservedthe spontaneous 4. Simple phenomenological formula for the
emission of 14C from 223Ra. The cluster emission is a spontaneous fission half-lives
very rare process. Its relative branching ratio to the
α-decay is of the order 10−10 to 10−17. Nevertheless, Encouraged by the good result for Tα obtained in
1/2
in lasttwodecades one has observedclusters from14C the Gamow theory, we are going in the following
to34Siemittedbyactinidenucleifrom221Frto242Cm. to describe in another simple model the spontaneous
Inallcasestheresidualnucleuswasalwaysclosetothe fission half-lives of transactinide nuclei. Namely, we
double magic 208Pb. shalladopttheS´wia¸teckiideafrom1955[9],whofound
a simple relationbetweenthe spontaneousfissionhalf-
lives and the experimental mass deviations from their
liquid drop estimates. The crucial ingredient of the
CE model is the liquid drop formula which is described in
MeV
the next subsection.
AF
4.1. The liquid drop model
In our analysis we decided to use the modern
version of the macroscopic-microscopic model [10].
The macroscopic part of this Lublin-Strasbourg Drop
(LSD) mass formula (in MeV units) is as follows:
M (Z,N,def)=
LSD
7.289034·Z+8.071431·N −0.00001433·Z2.39
−15.4920(1−1.8601I2)A
Figure 8. TheHFBpotential energyof230Thasafunctionof
quadrupoleandoctupolemoments. Thethethicksolidlineshow +16.9707(1−2.2938I2)A2/3Bsurf(def) (13)
thepathtoclusteremission(CE)whilethedashedoneisdrawn +3.8602(1+2.3764I2)A1/3B (def)
cur
alongthevalleytoasymmetricfission(AF).AfterRef.[30].
+0.70978Z2/A1/3B (def)−0.9181Z2/A
Coul
In general, there are two alternative theoretical −10exp(−4.2|I|)Bcong(def)+Eo−e .
On spontaneous fission and α-decay half-lives of atomic nuclei 8
where the odd-even energy term Ee−o is given in half-lives in years is given by:
Ref. [38]. In Eq. (13) A = Z + N denotes the
mass number, I = (N − Z)/A reduced isospin and log10 T1ys/f2 =−4.1·min(Z,103)+380.2
B , B , B and B are relative to the (cid:18) (cid:19)
surf cur Coul cong 0 for even−even , (16)
sphere: surface, curvature, Coulomb and congruence
−7.7δM(Z,N)+ 2.5 for odd−A ,
(see Ref. [38]) energies. The parameters in the first 
5 for odd−odd .

and the last row in Eq. (13) are taken from Ref. [39],
while the rest 8 parameter were fitted to the data. 
4.2. Simple model `a la S´wia¸tecki Th exp
20
th
Following the idea presented in Ref. [9], we are going
)
tloogfianridthamnsaopfprsopxoinmtaantievoeufsunficsstiioonnahladlef-plievnedse,nccoerroefctthede sf / y 1/2 10 UPu Ds
by mass shifts kδM, on proton Z and neutron N ( T 0 Cm No Sg
numbers: 0
1 Hs Fl
g
f(Z,N)=log [Tsf /y]+kδM(Z,N) . (14) o Cf
10 1/2 l -10
The δM is a ground-state microscopic energy, defined ( a ) Fm Cn
Rf
as: -20
δMexp (Z,N)=M (Z,N)−M (Z,N,0), (15) 34 36 38 40 42 44 46
micr exp LSD 2
Z / A
whereM (Z,N)isameasuredmassofisotope,taken
exp
from [40] and M (Z,N,0) were calculated using
LSD
formula (13). 20 Pa exp U
th
) 10 Am Cm Cn
1200 e-e sfT / y 1/2 -100 Es Lr Bh Fm Rf Hs
M o-A (
sfδT / y ) + k1/2 --21 000 o-o log 10 -1120000 BN(k pb )Md Db Pu CNfo Sg Ds
(
10 -30 36 38 40 42 44 36 38 40 42 44
g
lo -40 Z2 / A Z2 / A
-50 Figure 10. Spontaneous fission half-lives of even-even (a)
90 94 98 102 106 110 114 and odd (b) nuclei, calculated using formula(16) (triangles) in
Z comparisontotheexperimentalvalues [11](dots).
Figure 9. Logarithmsoftheobservedspontaneous fissionhalf- Spontaneous fission half-lives, calculated using
livescorrectedwithmasses”shifts”kδM asafunctionofproton formulaEq.(16)arepresentedandcomparedwith the
number. data in Fig. 10. Isotopes which for masses are still not
measuredaremarkedwithopensymbols. Averygood
Fitting procedures included 35 fissioning even-
accuracy was achieved in actinides region, where the
even isotopes (Z ≤ 102) with measured masses and
root-mean-squaredeviation of logTsf is equal to 1.18
half-lives. The smooth dependence for even-even 1/2
and it grows to 1.57 when odd-A and odd-odd nuclei
isotopeswasachievedforfactork =7.7MeV−1,asitis
are taken into account. The r.m.s. deviation reaches
showninFig.9. Thecurves,fittedforodd-Aandodd-
1.96,whenoneincludestotheanalysisthesuper-heavy
odd nuclei are shifted by a constant- hindrance factor.
isotopes, for which the liquid drop barrier vanishes.
It is equal to h = 2.5 for odd-A isotopes and doubled
Thequalityofreproductionofknownlife-timesbythis
for odd-odd systems. Formula for spontaneous fission
simple model is even better than that presented in
Fig. 6 obtained in the macroscopic-microscopic model
with the cranking inertia [18]. Also more advanced
modern calculations based on the HFB theory with
On spontaneous fission and α-decay half-lives of atomic nuclei 9
the Skyrme (see e.g. [41, 42] or Gogny force [43] theorem reallyworks[46]. All estimates do not exceed
did not bring better estimates for large sample of 0.67MeVandliebelowtheexperimentalvalues(except
isotopes than the simple formula presented above. It 250Cf), what give a place for not washed out shell
raises a question: why the simple S´wia¸tecki model effects around the saddle point energy.
works? To understand it, one has to recall S´wia¸tecki’s
topographical theorem and the results obtained using 4.4. Justification of the simple formula for T
sf
dynamic (least action) trajectories to fission described
Let us consider one dimensional fission barrier along
in Section 3.
the least-action (dynamic) trajectory described in
Sec.3. Assuming,thatthefissionpathisparametrised
4.3. The topographical theorem
by the collective coordinate s, one can evaluate
the potential V(s) and the mass parameter B (s)
ss
corresponding to this path. The fluctuations of the
exp inertia B along the least-action trajectory becomes
10 ss
V] LSD smaller and smaller, when one increase the number
Me 8 of collective coordinates. Especially the collective
Th
xp [s 6 U Cm pflauicrtiunagtidoengsreoefsthoef finreeerdtioamfusnigctniiofinca[4n7tl]y. wash out the
eg
M
- 4
d Pu Cf
d
a
s 2
M
V
sadd
0
232 236 240 244 248
V(x)
~
A x) V(x) V
( B
V
Figure11. Experimentalfissionbarrierheightscomparedwith
theirestimatesevaluatedaccordingtothetopographicaltheorem
ofS´wia¸teckiandtheLSDmacroscopicenergy. AfterRef.[46]. E
0
It was shown in Refs. [10, 27, 44] that the
LSD model (13), which parameters were fitted to
x 0 x
l r
the experimental ground-state masses only, is able
to reproduce well the fission barrier heights of light,
Figure 12. Schematic plot of a fission barrier in form of two
mediumandheavynuclei,whenthemicroscopicpartof
invertedparabolassmoothlyjoinedatthescissionpoint.
the ground-state binding energy is taken into account.
According to the topographical theorem, proposed by A simple transformation:
Myers and S´wia¸tecki [38], the mass of a nucleus in a
s
saddle-point is mostly determined by the macroscopic Bss(s′) ′
x(s)= ds , (18)
part of its binding energy. The influence of the m
Z r
shell effects on the saddle-point energy is rather weak ssadd
and in the first approximation might be neglected. bring us to the new coordinate x(s), in which the
So, one can evaluate the fission barrier height as a inertia becomes constant B = m. The lower
xx
differencebetweenthemacroscopic(hereLSD)andthe integration limit in Eq. (18) is chosen at point x = 0,
experimental mass: correspondingtothe saddlepoints . Thecollective
sadd
V (Z,N)=M (Z,N,saddle)−Mg.s., (17) potential in the new coordinate, schematically plotted
B LSD exp in Fig. 12, can be approximated by two inverted
whereMLSD(Z,N,saddle)isamacroscopicpartofLSD parabolas of different stiffnesses Cl and Cr having the
mass formula (13) taken in a saddle point. maximum at the saddle point:
Thecomparisonoftheexperimentalfissionbarrier
V − 1C x2 for x < 0 ,
heightofactinidenucleiwiththeirestimates,evaluated V(x)= sadd 2 l (19)
according to the topographical theorem of S´wia¸tecki ( Vsadd− 12Crx2 for x > 0 ,
using the LSD macroscopic energy, is presented in
e
The stiffnesses of the V potential are chosen in such a
Fig. 11. The r.m.s. deviation of the estimates from
the data for all 18 considered fission barriers heights
e
is 0.31 MeV only, what proves, that the topographical
On spontaneous fission and α-decay half-lives of atomic nuclei 10
way that the action-integral S (9) becomes equal:
x′r xr 20
2m 2m
[V(x)−E ]dx= [V(x)−E ]dx (20) M
−waZxnh′ldrerrei¯hghp2taitrusrn(−inxg′lp,xo0′rin)tsanfo−drZxtl(hr−ext¯hlr,2uxere)anadreapcplar0ossxiicmalatlievfet sfδ / y ) + k1/2 0 e-e
T -20 o-A
potentialrespectively. ThelastintegralinEq.(20)can (
0
be rewritten as g1 o-o
o
2m xl Clx2 xr Crx2 l -40 fe-e (VBLSD)
S = V − dx+ V − dx ,
r¯h2 Z0 r B 2 Z0 r B 2  0 1 2 3 4 5
LD
V [MeV]
B
where VB =Vsadd−E0 is the fission barrier height.
After a small algebra the action integral becomes
Figure13. Logarithmsoftheobservedspontaneousfissionhalf-
π m m π ω +ω
S = V + = V l r , (21) lives corrected with 7.7δM as a function of liquid drop barrier
B B
2h¯ C C ¯h 2ω ω height.
(cid:18)r l r r(cid:19) l r
where ω = C /m and ω = C /m are frequencies
l l r r
of the left and right (inverted) oscillator respectively. duetotheshelleffectsinthegroundstate. Thesmooth
p p
Introducing the average oscillator frequency dependence of logarithms of spontaneous fission half-
2ωω lives, corrected by the ground state shell-plus-pairing
l r
ω = , (22)
ω +ω effects, on the LSD fission barrier heights, shown in
l r
Fig. 13 for even-even (e-e), odd A (o-A) and odd-odd
onecanbringtheactionintegraltothefollowingform:
e (o-o) nuclei. The data for e-e isotopes lie very close
2V
S = B . (23) to the straight line, what validates Eq. (25). The data
¯hω
for o-A and o-o are above this line, what is due to the
In our approximation the action integral is propor- specializationenergy,whichincreasethe fissionbarrier
tional teo the fission barrier height measured in the en- heights. Asitwasmentionedbefore,stabilityofsuper-
ergy quanta of the harmonic oscillator, which approx- heavy nuclei is determined by shell effects. The liquid
imates the fission barrierform. For the action-integral drop barrier of these isotopes vanishes, what is visible
S > 1 the logarithm of the spontaneous fission half- in Fig. 13 for nuclei with Z ≥103.
lives (7) can written as
log(Tsf /y)=2S−log(n)−0.3665 (24) 5. Conclusions
1/2
where the constant 0.3665=log[ln(2)] and n is the
The following conclusions can be drawn from our in-
number of assaults of nucleus on the fission barrier
vestigation:
per year (y). Having in mind, that according to
• Simple,consistentmodelwasappliedtoreproduce
the topographical theorem the fission barrier height is
alpha decay half-lives in 360 nuclei with atomic
V = Msadd −Mgs and making use of the relation
B LSD exp number 52≤ Z ≤110.
(23), one can rewrite the last equation as follows:
• Model reproduces α−decay half-lives with quite
4δMexp 4VLSD
log(Tsf /y)+ micr = B −log(n)−0.3665,(25) good accuracy;the root-mean-squaredeviation of
1/2 ¯hω˜ ¯hω˜ log (Tα )foreven-evenisotopesisequalto0.39.
10 1/2
where δMmexipcr was defined in Eq. (15) and VBLSD = • Large underestimations in half-lives of N = 127
Msadd − Msph is the liquid drop fission barrier isotones arises from strong shell effects, which are
LSD LSD
height. The right hand side of Eq. (25) is a very not considered in this simple model.
smooth function of nucleon numbers as it is defined
• Semi-empiricalformulaforthespontaneousfission
only by global properties of nucleus. Note, that
half-lives, depending on proton number and the
the derived equation has the same structure as the
ground state microscopic corrections, reproduces
phenomenological formula of S´wia¸tecki (14), what
data for even-even super-heavy nuclei with rea-
proves his Ansatz.
sonable accuracy.
The liquid drop barrier height of actinides
• Quality of spontaneous fission half-lives evalua-
decreasesalmostlinearlyinfunctionofZ from4.3MeV
tionbreaksdownfornucleiwithnotmeasuredyet
for Z =90to 0 for Z ≥103 [44]. The fissionbarrierof
masses.
finite height appears in the super-heavy nuclei mostly